A weight that is attached to the end of a spring is pulled and then released. The function $H$ gives its height, in centimeters, after $t$ seconds. $\newline$ What is the best interpretation for the following statement? $\newline$ $H^{\prime}(0)=3$ $\newline$ Choose $1$ answer: $\newline$ (A) When the weight is released, its height is $3$ centimeters. $\newline$ (B) When the weight is released, its mass is increasing at a rate of $3$ grams per second. $\newline$ (C) When the weight is released, its height is increasing at a rate of $3$ . $\newline$ (D) When the weight is released, its height is increasing at a rate of $3$ centimeters per second.

Full solution

Q. A weight that is attached to the end of a spring is pulled and then released. The function

H

gives its height, in centimeters, after

t

seconds.

\newline

What is the best interpretation for the following statement?

\newline

H^{\prime}(0)=3

\newline

Choose

1

answer:

\newline

(A) When the weight is released, its height is

3

centimeters.

\newline

(B) When the weight is released, its mass is increasing at a rate of

3

grams per second.

\newline

3

\newline

(D) When the weight is released, its height is increasing at a rate of

3

centimeters per second.

Definition of Derivative: $H^{\prime}(0)=3$ means the derivative of the height function $H$ with respect to time $t$ is $3$ when $t$ equals $0$ .
Rate of Change Interpretation: The derivative represents the rate of change of the height with respect to time.
Initial Rate of Change: Since the derivative is taken at $t=0$ , it describes the initial rate of change of the height right after the weight is released.
Rate of Change at $t=0$ : The value of the derivative is $3$ , which means the height is increasing at a rate of $3$ centimeters per second at $t=0$ .